# Properties

 Label 2-165-1.1-c1-0-3 Degree $2$ Conductor $165$ Sign $-1$ Analytic cond. $1.31753$ Root an. cond. $1.14783$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.41·2-s − 3-s + 3.82·4-s − 5-s + 2.41·6-s + 0.828·7-s − 4.41·8-s + 9-s + 2.41·10-s − 11-s − 3.82·12-s − 5.65·13-s − 1.99·14-s + 15-s + 2.99·16-s − 1.17·17-s − 2.41·18-s − 6.82·19-s − 3.82·20-s − 0.828·21-s + 2.41·22-s − 4·23-s + 4.41·24-s + 25-s + 13.6·26-s − 27-s + 3.17·28-s + ⋯
 L(s)  = 1 − 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.447·5-s + 0.985·6-s + 0.313·7-s − 1.56·8-s + 0.333·9-s + 0.763·10-s − 0.301·11-s − 1.10·12-s − 1.56·13-s − 0.534·14-s + 0.258·15-s + 0.749·16-s − 0.284·17-s − 0.569·18-s − 1.56·19-s − 0.856·20-s − 0.180·21-s + 0.514·22-s − 0.834·23-s + 0.901·24-s + 0.200·25-s + 2.67·26-s − 0.192·27-s + 0.599·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$165$$    =    $$3 \cdot 5 \cdot 11$$ Sign: $-1$ Analytic conductor: $$1.31753$$ Root analytic conductor: $$1.14783$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{165} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 165,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + T$$
5 $$1 + T$$
11 $$1 + T$$
good2 $$1 + 2.41T + 2T^{2}$$
7 $$1 - 0.828T + 7T^{2}$$
13 $$1 + 5.65T + 13T^{2}$$
17 $$1 + 1.17T + 17T^{2}$$
19 $$1 + 6.82T + 19T^{2}$$
23 $$1 + 4T + 23T^{2}$$
29 $$1 + 4.82T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 11.6T + 37T^{2}$$
41 $$1 - 4.82T + 41T^{2}$$
43 $$1 + 8.82T + 43T^{2}$$
47 $$1 + 4T + 47T^{2}$$
53 $$1 - 9.31T + 53T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 + 11.6T + 61T^{2}$$
67 $$1 + 5.65T + 67T^{2}$$
71 $$1 - 2.34T + 71T^{2}$$
73 $$1 - 11.3T + 73T^{2}$$
79 $$1 - 8.48T + 79T^{2}$$
83 $$1 + 10T + 83T^{2}$$
89 $$1 - 3.65T + 89T^{2}$$
97 $$1 - 11.6T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$